A geometric uncertainty principle with an application to Pleijel's estimate

TL;DR

This paper introduces a geometric uncertainty principle for domain partitions, showing that either the asymmetry or deviation of elements is bounded away from zero, and applies this to improve bounds on Laplacian eigenfunction nodal domains.

Contribution

It establishes a new geometric uncertainty principle and applies it to enhance Pleijel's estimate on nodal domain counts, improving previous spectral partition bounds.

Findings

Average partition element has bounded asymmetry or deviation from smallest element.

Improved bound on the number of nodal domains of Laplacian eigenfunctions.

Enhanced spectral partition estimates.

Abstract

Consider partitions of an open, bounded domain in $\mathbb{R}^n$. Then an average element of the partition has either its Fraenkel asymmetry or its deviation from the smallest element in the partition bounded away from 0 by a universal constant. As an application, we give an (unspecified) improvement of Pleijel's estimate on the number of nodal domains of a Laplacian eigenfunction similar to recent work of Bourgain and improve a bound coming from spectral partition problems.